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First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Edit: Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota, which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.

First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Edit: Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota,¹ which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.

¹Finberg, David; Mainetti, Matteo; Rota, Gian-Carlo, The logic of commuting equivalence relations, Ursini, Aldo (ed.) et al., Logic and algebra. Proceedings of the international conference dedicated to the memory of Roberto Magari, April 26–30, 1994, Pontignano, Italy. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 180, 69-96 (1996). ZBL0862.03039, MR1404934.

First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Edit: Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota, which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.

First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Edit: Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota, which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.

First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's master's thesis on multiplicative ideal theory. Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Edit: the link originally given for this thesis no longer works, but in the meantime, Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota, which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.

First, an answer to Pete Clark's comment on the Chinese remainder theorem can be found in Floris Ernst's 2004 University of Otago Master's thesis Multiplicative ideal theory (pdf link). Prüfer domains are exactly those integral domains for which the ideal lattice is distributive, and Ernst indicates that these are exactly those integral domains "satisfying CRT": see section 3.3 for an explanation of what he means.

Edit: Pete has updated his notes on commutative algebra to give an account of the relevant facts (section 21). See also this discussion on CRT and distributive lattices at Mathematics StackExchange.

Second, I'd just like to underscore what is perhaps the heart of the article mentioned by Marko Amnell: what Rota calls "linear lattices" (lattices of commuting equivalence relations), the most important class of modular lattices. These include lattices of ideals, lattices of normal subgroups, and much more generally, lattices of algebraic congruence relations that arise for algebras of a Mal'cev theory, much studied by universal algebraists.

The point is that such lattices (and I repeat that ideal lattices are examples) have an incredibly rich structure: they are not just modular lattices, they are also Desarguesian (satisfy an axiom which generalizes the Desarguesian axiom for projective planes), and in fact satisfy a battery of equational identities incapable of being finitely axiomatized. I'm mentioning all this because I think it's highly likely that linear lattices were very much on Rota's mind at the time of writing Indiscrete Thoughts. See for example this paper from Google books, by Finberg, Mainetti, and Rota, which gives a kind of natural deduction calculus for linear lattices.

If it is indeed the case that Grothendieck was rediscovering some of the identities known to occur in lattices of commuting equivalence relations, then perhaps Rota would have a point there. But I'm not sure about that: for example, Rota credits Schützenberger with the discovery of the Desarguesian identity, and I think this might have occurred post 1965. I haven't looked into the history (nor do I have any but the first volume of EGA at hand; treillis wasn't in the index there).

I can't resist adding, for anyone who might be interested, that the "logic" of linear lattices developed by Rota and his collaborators smells very similar to the graphical calculus outlined in Categories, Allegories, by Freyd and Scedrov (section 2.158), for deciding which equational identities expressible in the language of allegories holds in the allegory of relations between sets. Yes, Freyd and Scedrov indicate that this theory is decidable (which seems related to concerns expressed by Rota in that AMS Notices article), and the decision algorithm is based on graphs which admit a "parallel-series" decomposition in the sense of circuit theory, which looks on the face of it very similar to the graphical calculus given by Finberg, Mainetti, and Rota. I don't know whether categorical logicians have been talking much with lattice theorists, even though they were some of both at the conference where Finberg, Mainetti, and Rota read their paper.